Learn Dutch: Ik ben vandaag jarig. - It is my birthday today.

In English, you use a noun phrase: "It is my birthday." In Dutch, having a birthday is expressed with the adjective jarig combined with the verb zijn (to be). You are literally saying "I am birthday-ing." That is why you use Ik ben.

Why does the time word vandaag come before jarig?

In Dutch word order, expressions of time like vandaag (today) usually come as close to the first verb as possible. It is much more natural to place the time right after ben and before the adjective jarig.

Since this lesson is about the future tense with gaan, how would I say "It is going to be my birthday tomorrow"?

You would use gaan (to go) and move the infinitive verb zijn (to be) to the very end of the sentence: Ik ga morgen jarig zijn. However, Dutch speakers frequently just use the present tense with a future time word, so Ik ben morgen jarig is also very common and correct.

How would I use this to talk about someone else's birthday, like "It is his birthday"?

You simply change the subject and conjugate the verb zijn to match. "It is his birthday" becomes Hij is jarig (literally: He is birthday-ing), and "It is my mother's birthday" becomes Mijn moeder is jarig.